Further Mathematics ยท WFM01 (F1)

Complex numbers: the basics

Complex numbers let every quadratic have roots. They are built on i, where i2 = -1.

The form a + bi

A complex number z = a + bi has real part a and imaginary part b. sqrt(-9) = 3i because (3i)2 = 9i2 = -9.

Arithmetic

Add and subtract real and imaginary parts separately. Multiply out brackets and replace i2 with -1.

(2 + 3i)(1 - i) = 2 - 2i + 3i - 3i2 = 2 + i + 3 = 5 + i.

The conjugate

The complex conjugate of z = a + bi is z* = a - bi. z z* = a2 + b2, a real number. To divide, multiply top and bottom by the conjugate of the bottom.

Worked example

Solve x2 + 4x + 13 = 0.

  1. x = (-4 +/- sqrt(16 - 52)) / 2 = (-4 +/- sqrt(-36)) / 2
  2. sqrt(-36) = 6i

Answer: x = -2 + 3i or x = -2 - 3i

Key idea: i2 = -1; z* = a - bi; z z* = a2 + b2; divide by multiplying by the conjugate.

Check yourself

  1. What is sqrt(-16)?
    • -4
    • 4i
    • 16i
    • -4i2
    Show answer

    4i. (4i)2 = 16i2 = -16.

  2. Simplify (1 + 2i)(3 - i).
    • 3 - 2i
    • 1 + 5i
    • 5 + 7i
    • 5 + 5i
    Show answer

    5 + 5i. 3 - i + 6i - 2i2 = 3 + 5i + 2 = 5 + 5i.

  3. What is the conjugate of 4 - 7i?
    • -4 + 7i
    • 4 + 7i
    • -4 - 7i
    • 7 - 4i
    Show answer

    4 + 7i. Change the sign of the imaginary part.

  4. z = 3 + 4i. Find z z*.
    Show answer

    25. z z* = 9 + 16 = 25.

  5. i2 = -1. Find i4.
    Show answer

    1. i4 = (i2)2 = (-1)2 = 1.

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Sources

Last updated 7 November 2026. Spotted a mistake? Email alevel@lifereadysociety.com.